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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 931 (2018) 216–225

www.elsevier.com/locate/nuclphysb

Noether symmetry approach in the cosmological

alpha-attractors

Narakorn Kaewkhao

, Thanyagamon Kanesom

Phongpichit Channuie

b,∗

Department of Physics, Faculty of Science, Prince of Songkla University, Hatyai 90112, Thailand

School of Science, Walailak University, Thasala, Nakhon Si Thammarat, 80160, Thailand

Received 19

January 2018; received in revised form 10 April 2018; accepted 16 April 2018

Available

online 21 April 2018

Editor: Stephan

Stieberger

Abstract

cosmological framework, Noether symmetry technique has revealed a useful tool in order to exam-

ine

exact solutions. In this work, we ﬁrst introduce the Jordan-frame Lagrangian and apply the conformal

transformation in order to obtain the Lagrangian equivalent to Einstein-frame form. We then analyze the

dynamics of the ﬁeld in the cosmological alpha-attractors using the Noether symmetry approach by focus-

ing

on the single ﬁeld scenario in the Einstein-frame form. We show that with a Noether symmetry the

corresponding dynamical system can be completely integrated and the potential exhibited by the symmetry

can be exactly obtained. With the proper choice of parameters, the behavior of the scale factor displays an

exponential (de Sitter) behavior at the present epoch. Moreover, we discover that the Hubble parameters

strongly depends on the initial values of parameters exhibited by the Noether symmetry. Interestingly, it

can retardedly evolve and becomes a constant in the present epoch in all cases.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

Corresponding author.

E-mail

addresses: naragorn .k @psu .ac .th (N. Kaewkhao), thanyagamon1995 @gmail .com (T. Kanesom),

channuie @gmail .com (P. Channuie).

https://doi.org/10.1016/j.nuclphysb.2018.04.011

The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

N. Kaewkhao et al. / Nuclear Physics B 931 (2018) 216–225 217

1. Introduction

In modern cosmology, the mechanism of cosmic inﬂation seems conceivable. Inﬂation marks

nowadays an inevitable ingredient when studying very early evolution of the universe. The reason

stems from the fact that it solves most of the puzzles that plague the standard Big Bang theory,

and simultaneously is consistent with recent observational data. In other wo

rds, it not only gives

sensible explanations for the horizon, ﬂatness, and relic abundant problems, but also provides

us primordial density perturbation as seeds of the formation for a large-scale structure in the

universe.

Despite all its success, ho

wever, the underlying mechanism of the inﬂationary physics is still

unknown. Recent observational data much ﬂavors large ﬁeld inﬂationary models with plateau-

like inﬂaton potentials. However, there are several different ways of constructing successful

inﬂationary models. In the supergravity context, a model with plateau potentials was proposed

by the authors of Refs.

[1,2]. Here it describes a potential which is exponentially approaching a

positive constant for super-Planckian values of the inﬂaton ﬁeld. Later on, a theory with a similar

potential was realized as the Starobinsky model [3], and then the Higgs inﬂation model with a

similar potential wa

s developed [4,5]. It is worth noting that these models lead to nearly identical

predictions, providing the best ﬁt to the latest Planck data [6,7]. Moreover, recent investigation

shows the inﬂaton ﬁeld can emerge as a composite state of a new strongly interacting gauge

theory [8,9]. More recently

, a broad class of inﬂationary models, dubbed cosmological attractors

[10–16], yields very similar inﬂationary predictions.

Interestingly, the cosmological α-attractors incorporate most of the e

xisting inﬂationary mod-

els with plateau-like potentials including the Starobinsky model and some generalized versions

of the Higgs inﬂation. Regarding the α-attractors, the ﬂatness of the inﬂaton potential is imple-

mented and protected by the existence of a pole in the kinetic term of the scalar ﬁeld. M

oreover,

at large-ﬁeld values, any non-singular inﬂaton potential acquires a universal plateau-like form

when performing the (conformal) transformation. Regarding the hyperbolic geometry and the

ﬂatness of the Kahler potential in the supergravity context, the universal behaviors of these

theories make very similar cosmological predictions preserving good match to the latest cos-

mological observ

ations [17–20]. The successful extensions of these models can also describe

dark energy/cosmological constant and supersymmetry breaking [21–29].

The purpose of the present study is to analyze

the dynamics of the ﬁeld in the cosmological

α-attractors through the Noether symmetry technique. Here we concentrate on the single ﬁeld

scenario. It is worth noting that this approach proved to be very useful not only to ﬁx physi-

cally viable cosmological models with respect to the conserv

ed quantities, e.g., couplings and

potentials, but also to reduce dynamics and achieve exact solutions. Moreover, the existence of

Noether symmetries plays crucial roles when studying quantum cosmology [30].

In the present paper

, we consider the single-ﬁeld model of cosmological α-attractors. The

structure of the paper is as follows: In Sec. 2, we introduce the model in the Jordan frame (JF)

and transform the original action into the equivalent Einstein frame (EF) by applying the con-

formal transformation. In Sec. 3, we adopt the Noether symmetry approach to study the related

dynamical systems obtained from a point-lik

e Lagrangian. Here we can determine the form of

the undeﬁned potential of the action by imposing the Noether symmetry. We also compute the

general solutions with the helps of the new coordinate system. M

oreover, we compute the Hubble

parameter and display plot as a function of time for some speciﬁc cases. Finally, we conclude

our ﬁndings in the last section.

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